# Finding Shortest Path in the Plane with Obstacles

[CG]Maxime
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Among the games I have developed on CodinGame, one of my favorites is Ghost in the Cell because it gave me the opportunity to solve a very interesting problem: finding the shortest path in a plane while avoiding obstacles. This is an algorithm I've been taught during a robot motion planning class when I was a student but I never had the opportunity to implement it.

In this article, I will show you how to compute a path from one factory to another while avoiding other factories. A factory is simply a round obstacle.

The algorithm will be divided in 3 steps:

## Computation of the Polygons

The obstacles are round, however we need a discrete amount of vertices to build the visibility graph. A circle can easily be approximated by a regular polygon.

First, some vocabulary:

• A regular polygon is a polygon equiangular and equilateral
• The apothem ($a$) is the distance from the center to the midpoint of one of its sides
• The circumradius ($R$) is the distance from the center to one of its vertices
• The inscribed circle of a regular polygon is a circle with the same center and radius $a$

We are looking for a polygon for which the inscribed circle is the radius of a factory, plus a gap to forbid a path to go too close to an obstacle (too dangerous). The apothem of the polygon is : $a = obstacle.radius + extraSpace$. The circumradius can be calculated from the apothem and the number of edges: $R = \frac{a}{cos(\pi / n)}$.

The coordinates of the vertices of a polygon of $n$ edges are:

$\left \\left\{ \begin\left\{matrix\right\} x = obstacle.x + R \cdot cos\left(t\right) \\ y = obstacle.y + R \cdot sin\left(t\right) \end\left\{matrix\right\} \right .\quad\text\left\{with\right\}\quad t \in \left[0, \frac\left\{2\pi\right\}\left\{n\right\}, 2\frac\left\{2\pi\right\}\left\{n\right\}, 3\frac\left\{2\pi\right\}\left\{n\right\}, ..., \left(n - 1\right)\frac\left\{2\pi\right\}\left\{n\right\}\right]$

Here's the implementation of the polygon function:

Feel free to play with the number of edges.

## Visibility Graph

From the set of vertices of all the polygons, we create a graph called Visibility Graph. Each vertex is connected to all the vertices that can be reached by a direct line without crossing any obstacle.

For this, we need a function to detect an intersection between two segments. We'll also use a function to detect an intersection with a circle but it was not strictly necessary (this simplifies the code). The function directPath returns true when two points can be connected without intersecting a polygon.

The function buildGraph iterates over all the vertices of all the polygons and create a bidirectional edge when directPath returns true. We also add a unidirectional edge from each center (these edges only allows to exit a factory).

## Shortest Path

Any conventional shortest path algorithm can be used to compute the path from a factory to another using the visibility graph. Here's the implementation of an A*. For the sake of simplicity, I've used a sorted array instead of a priority queue, feel free to improve that part.

By increasing the number of edges for each polygon, the path become smoother.